3.1. Measurement of hydrogen moles permeated by volumetric analysis

The molar amount of hydrogen gas permeated from the polymer after high-pressure hydrogen injection is measured in real time by utilizing the permeation cell and the volumetric gas collection method^77-79 in graduated cylinder, as shown in Fig. 1, with the change in the water level being analyzed through an image brightness analysis program. As the hydrogen gas is diffused and permeated across the specimen, it displaces the water inside the cylinder, causing the water level on the Fig. 1(b) to gradually decrease, forming a crescent-shaped meniscus (denoted as h). According to the manometer principle, the pressure inside the empty space of the cylinder, P(t), as a function of time after pressure injection, can be expressed as the difference between the atmospheric pressure and the hydrostatic pressure, ρgh(t).^80-82

P(t) = P_o(t) – ρgh t (1)

Here, P(t) represents the atmospheric pressure in cylinder, ρ is the density of water at room temperature, g is the gravitational acceleration, and h(t) is the measured water level inside the cylinder, taken from the surface of the water container as a function of time. By measuring h(t), the volume of hydrogen gas (ΔV) permeated from the sample in the permeation cell can be determined. Using the ideal gas law (PV = nRT), the number of moles of hydrogen gas permeated (Δn) can then be calculated as follows.^81,82

Δn [mol] = P(t)ΔV(t)/RT(t) = [P_o(t) − ρgh(t)]ΔV(t) /RT(t), ΔV(t) = Ah(t) (2)

Here, R is the gas constant (8.20544 × 10⁻⁵ m³·atm/ (mol·K)), T(t) is the temperature inside the cylinder as a function of time, and A is the inner crosssectional area of the cylinder. The molar quantity of hydrogen gas permeated, based on the change in water level due to the hydrogen permeated from the sample, is obtained from Eq. (2).

3.2. Overall system for measuring hydrogen diffusivity, permeability and solubility

Fig. 1 shows the developed in-situ H₂ permeation measuring system, which allows for water level measurements caused by hydrogen permeated from specimen. This system consists of high pressure hydrogen tank, permeation cell, graduated cylinder, one temperature sensor (UA10, DEKIST Co., Ltd., Yongin, Korea), one pressure sensor (UA52, DEKIST Co., Ltd.), a digital camera (D800, Nikon Co., Tokyo, Japan) and a computer running the developed program. The temperature and pressure obtained from the USBbased temperature and pressure sensors were used to calculate the hydrogen mole penetrated from the sample using Eqs. (1) and (2).

The high-pressure hydrogen injection process using the system in Fig. 1 is as follows.

First, Valve 2 is closed and Valve 1 is opened to pressurize hydrogen up to 10 MPa, considering the maximum storage pressure of the hydrogen tank (12 MPa), and store it in the buffer tank. Next, Valve 1 is closed and Valve 2 is opened, allowing hydrogen to be injected into the permeation cell up to the target pressure within a few second. The measurement begins by recording the time from the moment when Valve 2 is opened (t = 0). The quantitative measurement of permeated hydrogen was conducted using the volumetric measurement with cylinder shown in Fig. 1(b). The partially submerged cylinders, filled with distilled water, were used to measure the changes in water level caused by the hydrogen permeated from the sample.

3.3. Development of H2 diffusion-permeation analysis program

Using the permeation measuring system in Fig. 1, the permeated hydrogen molar amount from the specimen were measured. The number of moles of the diffusing gas per unit area (A), Q(t), can be expressed as follows⁸³:

C₁ is the concentration at x = 0 in the high-pressure region. Here, x represents the spatial coordinate along the thickness of the specimen, where x = 0 corresponds to the high-pressure surface and x = l to the lowpressure (permeate) surface. D is diffusivity. According to the experimental setup in the permeation cell, for a specimen with thickness l, the polymer specimen initially had zero concentration and the concentration at the face (x = l) through which the gas diffused was effectively maintained at zero concentration. Thus, we have finally obtained the Eq. (3).

The infinite series expansion in Eq. (3) was computed by summing the first 50 terms, as higher-order terms were negligible, with values smaller than 10⁻⁵ . ^28,84 To ensure accuracy, we developed a specialized diffusionpermeation analysis program to include these 50 terms and calculate the diffusion coefficient using Eq. (3). This method proved to be more precise than the time lag (L) method, which is based on a simplified equation at infinite time, L = l ² /6D. ⁸³

The permeability was derived from the linear slope (Δn/Δt) representing the change in moles of hydrogen over time as follows:

P =(Δn ⁄Δt)l⁄AΔP (4)

Here, A denotes the hydrogen contact area of the specimen, and ΔP is the pressure difference between the feed and permeate sides in the permeation cell. l is the specimen thickness. The hydrogen permeability was then determined from the steady-state flow rate using Eq. (4). Meanwhile, the H₂ solubility (S) is obtained from the diffusivity (D) and permeability (P) as follows^32,36:

S = P/D (5)

Fig. 2(a) shows an example of analyzing the hydrogen diffusion coefficient (D) and permeability (P) of an EPDM polymer using a self-developed hydrogen diffusion-permeability analysis program, based on molar amount of hydrogen measured at a hydrogen injection pressure of 10.0 MPa. The molar amount of hydrogen permeated at each time is obtained from Eq. (2). The analysis process is as follows:

First, after entering the thickness, effective permeation area (Area), experimental pressure difference (ΔP) and H₂ measured molar amount data for the diskshaped specimen in the lower-left section, the curve fitting function, located in the middle-right section, is executed. As a result, the lower-right section displays the diffusivity, permeability and solubility values calculated using Eqs. (3) − (5), yielding D = 2.172 × 10⁻¹⁰ m²/s, P_e = 3.590 × 10⁻⁹ mol/m·s·MPa and S = 16.529 mol/m³ ·MPa, respectively. Utilizing this dedicated analysis program significantly reduces measurement time and provides additional information, such as the Figure of Merit (FOM) = 0.9 %, which indicates the deviation between equations and the measured data, as shown in the middle section of Fig. 2. The Fig. 2(b) is replotted results with permeation parameters obtained by diffusion-permeation analysis program. The blue line is the fitted result based on Eq. (3) by analysis program.

4.1. Hydrogen permeation parameter versus injection pressure

Using a self-developed high-pressure hydrogen permeation system, the hydrogen permeation properties of EPDM polymer containing carbon black filler, designed for high-pressure hydrogen environments, were evaluated under various pressure conditions ranging from 1 MPa to 10 MPa. Fig. 3 presents the analysis of pressure dependence on permeation rate, permeability, diffusivity and solubility.

In Fig. 3(a), the hydrogen permeation rate exhibited an increasing trend with rising pressure. However, the rate of increase gradually diminished at higher pressures. Theoretically, if permeability remains constant regardless of pressure, Eq. (4) suggests that the permeation rate should form a linear relationship passing through the origin. However, the observed decline in the rate of increase indicates that permeability decreases as pressure rises, as shown in Fig. 3(b). This behavior aligns with previous studies on the pressure dependence of gas permeability in polymers. According to Stern et al.⁸⁵ and Naito et al. ⁸⁶ gas permeation in polymers can be classified into two categories based on the correlation between permeability and pressure. Condensable vapors, such as CO₂, N₂O, and organic vapors, exhibit a positive slope where permeability increases with pressure. In contrast, permanent gases with extremely low liquefaction temperatures, such as N2 and He, show a negative slope, indicating a decrease in permeability with increasing pressure. Since hydrogen is also classified as a permanent gas, our results in Fig. 3(b) confirm a similar trend, where permeability decreases as pressure increases.

Fig. 3(c) illustrates the pressure dependence of diffusivity, which also decreases with increasing pressure. Gas diffusion in polymers follows the free volume model, where gas molecules migrate through transient free spaces generated by the thermal motion of polymer chains.87,88 Fujiwara et al. 22 reported that, under high-pressure conditions, the available free volume within the polymer is reduced, limiting molecular diffusion and consequently lowering diffusivity.

In contrast, Fig. 3(d) shows that solubility remains relatively constant within the margin of uncertainty, regardless of pressure. This suggests that the decrease in permeability with increasing pressures is primarily attributed to reduced diffusivity due to the decrease in free volume, rather than changes in solubility. Additionally, the permeation behavior of hydrogen in polymeric materials under high-pressure conditions is influenced not only by the simple dissolution-diffusion mechanism but also by complex factors such as internal pore structures, pressure-induced structural changes, and interactions between the polymer matrix and fillers.⁸⁹

4.2. Hydrogen uptake versus injection pressure

Using the measured solubility values, the hydrogen uptake can be calculated by applying the following equation³²:

Hydrogen Uptake [wtppm] = S · m_H2 · P/d_s (6)

where m_H2 is the molecular weight of hydrogen gas (2.016 g/mol), P is the experimental pressure and d_s [g/m³] is the density of the specimen. The H₂ uptake was calculated using Eq. (6) for all solubility values presented in Fig. 3(d) and the results are shown in Fig. 4.

The behavior of hydrogen dissolved in polymeric materials can be generally interpreted using Henry’s law or the Langmuir adsorption model. Henry’s law describes the proportional increase in gas uptake with pressure, which is applicable when hydrogen is uniformly absorbed within the rubber matrix.⁹⁰ Meanwhile, the Langmuir adsorption model explains the phenomenon where hydrogen adsorbed onto the rubber matrix or filler surfaces reaches saturation at a certain pressure, preventing further adsorption.⁹¹ As shown in Fig. 4, the hydrogen uptake results for EPDM polymer containing carbon black filler followed Henry’s law with a squared correlation coefficient of R² = 0.98, indicating a quite good fit.

4.3. Uncertainty analysis

We have estimated the expanded relative uncertainty by finding the individual uncertainty factors. The uncertainty analysis was usually found in other literatures.^92-94 Table 2 presents the individual uncertainty factors and expanded uncertainties for H₂ permeability in our work. The primary sources of uncertainty in the permeability measurements were the repeated measurements and variations in the permeation area exposed to H₂ and in the sample thickness. The type A uncertainty for repeated permeability measurements was calculated based on five measurements. All type B uncertainties were determined using a rectangular distribution, dividing the uncertainty by \(\begin{align}\sqrt {3}\end{align}\) .

The accuracy of the graduated cylinder is 0.5 %, and applying a rectangular distribution, the type B uncertainty is calculated as 0.3 %. For a 10 mL graduated cylinder with a minimum scale division of 0.1 mL, the corresponding uncertainty is 1 %. Since the resolution is half of this minimum value, applying a triangular distribution of dividing factor (\(\begin{align}\sqrt{6}\end{align}\) ), the type B uncertainty is determined to be 0.2 %. The uncertainty in the measured volume of the permeated side of the permeation cell was determined to be 1.0 %, derived from the standard deviation of the data.

Regarding the permeation area, the designed area of the permeation cell was 962 mm² , but the O-ring seal caused a maximum area change of 48 mm² . This led to a variation in the permeation area of up to 5 %, with a corresponding type B uncertainty of 2.9 %, calculated from the rectangular distribution. The standard deviation between the measured data and the fitting results using Eq. (3) was within 0.9 %, and this value was used to determine the type B uncertainty. The uncertainty in sample thickness measurements was 0.8 %, based on the calibration certificate, accuracy and resolution of the Vernier caliper. After mounting the sample between the holders, the maximum change in thickness was 2 %, leading to a type B uncertainty of 1.2 % for the change in thickness, considering the rectangular distribution by dividing factor \(\begin{align}\sqrt{3}\end{align}\)

The combined standard uncertainty was obtained as the root sum of squares of individual uncertainty, assuming the individual uncertainty was independent of one another. To obtain the relative expanded uncertainty, the combined standard uncertainty was multiplied by a coverage factor of 2.1 at 95 % confidence level, under the assumption of a normal distribution. The resulting expanded uncertainty for the H₂ permeability was 9.1 % as detailed in Table 2.